Intro to Systems of Equations

Intro to Systems of Equations

First, we start with simple equations.

• A linear equation can be written in two formats:

• STANDARD: Ax+By=C, where SLOPE is -A/B, and y-intercept= C/B

• SLOPE-INTERCEPT: y=mx+b, where m=slope and b=y-intercept

Multiple Choice

y=2x-10. What is the slope, what is the y-intercept?

• slope=2, y-intercept=-10

• slope=-10, y-intercept=2

Multiple Choice

In y=2x-10, what does the slope of 2 mean?

• the line passes through (0,2)

• When x increases by 1, y increases by 2.

• Absolutely nothing

Multiple Choice

In y=2x-10, what does the y-intercept of -10 mean?

• the line passes through (0,-10)

• When x increases by 1, y decreases by 10

• absolutely nothing

Multiple Choice

How many possible solutions does y=2x-10 have?

• zero

• one

• two

• infinity

A linear equation in the form y=mx+b has infinity solutions

• As long as there are no restrictions (or limits) on what x-values you are using, y can be any real number from  $-\infty$  to  $+\infty$  

• Pick any real number you want!  Multiply it by m (slope), and then add the y-intercept (b).  Don't be afraid!

Multiple Choice

Which of these equations has the same slope as y=2x-10?

• y=5x-10

• y=2

• y=2x+5

• 2y=2x-10

What happens when two lines have the same slope but different y-intercepts?

• Let us examine the lines y=2x-10 and y=2x+5

• Both lines have the same steepness. When x increases by 1, y increases by 2

• However, the first line passes through (0,-10) and the second line passes through (0,5).

The blue line is y=2x+5; the red line is y=2x-10

• NOTICE: the two lines look like railroad tracks. The word for this is "parallel".

• RULE: When two lines are parallel, they both go forever in the same direction. They never intersect.

• There is no pair of (x,y) coordinates shared by these lines.

Multiple Choice

True or False: Two lines with the same slope NEVER intersect

• True

• False

Let us work it out algebraicly.

We see that if we set 2X-5 equal to 2X+10, we would normally solve by subtracting 2X from both sides.

When we do that, we get -5=10. That answer is NOT TRUE.

Therefore, this system of equations has NO SOLUTION, because there is no point where they intersect.

Open Ended

Let's take the system:

y=5X + 10

X=10.

What is the solution to this system of equations?

BREAK IT DOWN: Y=5X+10; X=10

We use substitution. Replace X with 10.

Y=5(10) + 10. We get Y=50+10. So the solution is (10,60).

IF we graph both lines

The red line is y=5X+10. We see that when X increases by 1, y increases by 5.

The green line is x=10. It is a straight vertical line that passes through (10,0).

The two lines intersect at (10,60).

Open Ended

2X + Y = 100

X=15

what is the solution to this system of equations?

BREAK IT DOWN: 2X+Y=100; X=15

First, let us substitute 15 for X

2(15) + Y = 100, so 30+Y=100.

Subtract 30 from both sides, Y=70

The solution is (15,70)

IF WE GRAPH the lines

The green line is 2X+Y=100.

The purple line is X=15.

The two lines intersect at (15,70)

Open Ended

y=4X+10; y=2x.

What is the solution to this system of equations?

THIS ONE WAS HARDER. BREAK IT DOWN

y=4x+10; y=2x. For this, we need to substitute 2x for Y.

This gives us 2x = 4x + 10

This one is tricky

SUBTRACT 4X from both sides. You get -2X=10. Divide both sides by -2, Y=-5

Now to solve for X

We said Y=2x. 2(-5)= -10

Therefore, the solution is (-5,-10)

USING A GRAPH

The green line is y=4x+10, the purple line is y=2x

The two intersect at (-5,-10). That is the solution.

Multiple Choice

LAST ONE FOR TODAY:

x+y=5

2x+2y=10

• This has no solution

• This has infinite solutions

• (5,10)

CAREFUL NOW. trick question!!!

You see that when we do this, the solution is 10=10. Of course, we already know that.

When you get an answer like this, the solution is "infinity"

Multiple Choice

JUST FOR FUN: What country is Godzilla from?

• China

• Japan

• South Korea

• USA

Multiple Choice

JUST FOR FUN: What anime is about a gun slinging bounty hunter from outer space?

• Dragon Ball Z

• Higorashi no naku koro ni

• Ready or Die

• Cowboy Bebop

Multiple Choice

JUST FOR FUN: What is another word for ninja (commonly used in "Naruto")

• Genjutsu

• Shinobi

• Arigato

• Kamikaze